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We establish the asymptotic theory in quantile autoregression when the modelparameter is specified with respect to moderate deviations from the unitboundary of the form (1 + c / k) with a convergence sequence that diverges at arate slower than the sample size n. Then, extending the framework proposed byPhillips and Magdalinos (2007), we consider the limit theory for thenear-stationary and the near-explosive cases when the model is estimated with aconditional quantile specification function and model parameters arequantile-dependent. Additionally, a Bahadur-type representation and limitingdistributions based on the M-estimators of the model parameters are derived.Specifically, we show that the serial correlation coefficient converges indistribution to a ratio of two independent random variables. Monte Carlosimulations illustrate the finite-sample performance of the estimationprocedure under investigation.

Asymptotic Theory for Unit Root Moderate Deviations in Quantile Autoregressions and Predictive Regressions